The center of pressure (CoP)

Consider a rigid body in contact with the ground plane. Let the ground plane normal be \(n\). Let various contact forces \(F_i\) act on the rigid body at positions \(r_i\). The center of pressure is then defined as
\begin{equation}
x_c = \frac{\sum_i (F_i \cdot n) r_i}{\sum_i F_i \cdot n}
\end{equation}
where the generalization from finitely many forces to a force field should be obvious. We generally assume that the normal component of the contact forces, \(F_i \cdot n\), is greater or equal to 0 (i.e. that the contact forces are non-sticky). This implies that, for
\begin{equation}
\alpha_i := \frac{F_i \cdot n}{\sum_j F_j \cdot n}
\end{equation}
we have \(0 \le \alpha_i \le 1\). The center of pressure, as defined above, is thus a convex sum and must lie within the convex hull of the contact points \(r_i\).